Real Analytic Functions#

Definition: Real Analytic Function

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \(O\) be an open subset of \(\mathbb{R}\) which is contained in \(\mathcal{D}\).

We say that \(f\) is analytic on \(O\) if there exists a real power series \(\displaystyle \sum_{n=0}^\infty a_n (x - c)^n\) which converges on \(O\) such that

\[ f(x) = \displaystyle \sum_{n=0}^\infty a_n (x - c)^n \qquad \forall x \in O \]

Theorem: Differentiation of Real Analytic Functions

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \(O\) be an open subset of \(\mathbb{R}\) which is contained in \(\mathcal{D}\).

If \(f\) is analytic on \(O\) with \(f(x) = \sum_{n=0}^\infty a_n (x - c)^n\), then \(f\) is differentiable on \(O\) and its derivative is also analytic on \(O\) with

\[ f'(x) = \sum_{n=1}^\infty n a_n (x - c)^{n-1} \]

Proof

TODO

Theorem: Antidifferentiation of Real Analytic Functions

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real function and let \(O\) be an open subset of \(\mathbb{R}\) which is contained in \(\mathcal{D}\).

If \(f\) is analytic on \(O\) with \(f(x) = \sum_{n=0}^\infty a_n (x - c)^n\), then its antiderivatives are also analytic on \(O\) with

\[ \int f(x) \mathop{\mathrm{d}x} = \text{const} + \sum_{n = 0}^\infty a_n \frac{(x - c)^{n+1}}{n+1} \]

Proof

TODO

Theorem: Automatic Differentiation via Dual Numbers

Let \(f: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}\) be a real analytic function and let \(a + b\varepsilon\) be a dual number.

If \(a \in \mathcal{D}\), then

\[ f(a + b\varepsilon) = f(a) + b f'(a)\varepsilon \]

Tip: Automatic Differentiation

This theorem allows us to find the derivative of \(f\) at any \(a \in \mathcal{D}\) without the need to find an expression for \(f'\), so long as we have a closed-form expression for \(f\). All we have to do is set \(b = 1\) and then use this expression to evaluate \(f(a + b \varepsilon)\). In the end, the value of the derivative \(f'(a)\) will be the coefficient before \(\varepsilon\).

Proof

TODO